It has been a long time that I haven’t updated this blog. I promised to post my readings on the Bennett-Bez-Carbery-Hundertmark’s short but beautiful paper, “heat flow monotonicity of Strichartz norm”. Here it goes!

In this paper, the authors applied the method of heat-flow in the setting of Strichartz inequality for the Schr\”odinger equation and amusingly obtained, among other things, the Strichartz norm

is nondecreasing as the initial datum evolvs under a under quadratic flow when . This immediately yields that the Gaussians are extremisers to the classical Strichartz inequality,

The paper also consider the monotonicity given by some other flows such as Mehler-flow; it also considers the higher dimensions’s analogues by embedding the usual Strichartz norm into one-parameter family of norms . In this post I will focus on the monotonicity induced by the heat-flow.

The method of heat-flow deformation is interesting in itself. It is expected to be applied to related problems such as the existence of extremisers to the adjoint Fourier restriction operators to the sphere; but it is a different story; so far I can see an immediate difficulty in the latter setting, i.e., an representation formula for where is the standard surface measure for the unit sphere in .

Fortunately the representation formula of this kind is available in the paraboloid setting thanks to Hundertmark-Zharnitsky’s work; I already reported it in my previous post but I would like to record it here again basically because an “equation” is rare in analysis: for nonnegative ,

where is the projection operator onto the subspace of functions on which are invariant under the isometries which fix the direction . ( We have a similiar representation formula when . )

The paper innovatively combines this formula with monotonicity given by some heat-flow version of Cauchy-Schwartz inequality together to prove the following main theorem.

Let If is Schr\”odinger admissible and is an even integer which divides then the one-parameter family of functions

is nondecreasing for all ; i.e., is nondecreasing in the case and . Next we show how to prove this theorem in the case .

Given and nonnegative integrable functions on , we define . Then is nondecreasing for all . Instead of repeating the proof in the paper of using the convolution with the heat-kernel, we will try working out how it is deduced from the divergence theorem. Our goal is to prove the following explicit formula

Let for . Then

We write . Then

. Then

.

Hence

. This is not surprising because formally solves the heat equation.

Let , then

.

Hence we see

Here we have used,

, which yields

,

i.e.,

We start with a general representation formula for the projection map in . For functions in we write

where is the group of isometries on which fix the direction , and denotes the right-invariant Haar probability measure on .

Let and . Then by using the fact the heat-flow operator commutes with tensor product and Hundertmark-Zharnistsky representation formula, we have

This is in form of . Then the monotonicity of follows from that of and the non-negativity of the measure .

For , define with , and the rescaled

Then

This leads to

On the other hand,

We observe that, by Plancherel theorem in both variables,

where denotes the surface measure of the paraboloid in . Hence we obtain

This shows that is an extremiser. So we are done.

You can see that the heat-flow deformation method is a very nice method; it has been applied by the authors in several settings such as the Hausdorff-Young inequality, and Young’s inequality (but I haven’t carefully read their previous works). It is also proved effective in treating -linear analogues of the Strichartz estimate (Bennett-Carbery-Tao’s work) and in the setting of multilinear Brascamp_Lieb inequalities (Bennett-Carbery-Christ-Tao) (I admitted that I haven’t read them carefully either). The “disadvantage” is that this method doesn’t charaterize the set of extremisers. Of course, we can’t hope it is a tool of catch-all. The question of charaterization is a completely different problem.

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